from what i know, there are 5 types of monsters
Agressive
Passive
Non-Agressive
Assist
Skill switch

Agressive attacks anything in sight
Passive just stands there not doing anything(with one exception)
Non Agressive does not attack until attacked back
Assist Attacks if one of its friends are attacked on the screen/closer
Skill Switch Monster switches target to whomever casts a spell on them